## Introduction

Some polymers have a yield stress that is linearly dependent on the logarithm of the strain rate, for some other polymers the yield stress is bi-linearly dependent on the log of the strain rate. This difference can be very important when trying to decide if you should perform high strain rate testing. In this article I will show how DMA temperature sweep data (storage and loss modulus) can help determine the strain-rate dependence of the yield stress. In short, it all depends on the beta-transition of the polymer.

## Yield Stress vs Strain Rate

Figure 1: Thermoplastic copolyester (Arnitel®).

If the **yield stress is ****linearly**** dependent on the applied log strain rate** then we only need to experimentally test the material at two different strain rates, and we can extrapolate to high and low rates. This is quite nice!

Figure 2: Experimental data for PVDF.

If the **yield stress is ****bi-linearly**** dependent on the applied log strain rate** then we need to experimentally test the material at both high and low strain rates.

## Polycarbonate (PC)

The following images show the DMA temperature sweep response of a polycarbonate (PC).

The image below shows that the yield stress of PC has a bi-linear dependence on the log strain rate. It is also clear that the material response becomes stiffer at low temperatures and at high strain rates, and similarly, the response becomes softer at high temperatures and low strain rates. This is a type of time-temperature equivalence. **It is also clear that the yield stress is bi-linear with respect to the log strain rate due to the beta-transition!**

## Thermoplastic Copolyester (Arnitel®)

This thermoplastic copolyester has a secant stress that is proportional to the log of the strain rate, and the for temperatures above 0°C the storage modulus is linearly dependent on the temperature. In this linear range the temperature and strain-rates are related by: \( T = T_0 + A (\log \dot{\varepsilon}_0 – \log \dot{\varepsilon})\). The DMA results, however, show that at really low temperatures the storage modulus starts to rapidly increase. This suggests that at sufficiently high strain rates the secant stress will be bi-linearly dependent on the log strain rate.

## Polyvinylidene Fluoride (PVDF)

This final example shows the behavior of PVDF. The yield stress is bilinearly dependent on the log strain rate due to the beta transition at low temperatures.

One way to write the time-temperature equivalence is to use the WLF equation. It is not clear, however, what the WLF parameters should be in this case. So I simply selected parameters makes the temperature to strain-rate mapping work: C1=25, C2=500 K, T0=48 K. The results are shown in the figure below. The red dots are the experimental yield stress values, and the blue line is the predictions from mapping the storage modulus vs temperature data. This is not a proof that this approach works, but it is cool that the results good!

## Summary

- For some polymers the yield stress is linearly dependent on the strain rate, for other polymers the response is bi-linear
- High strain rate testing is only needed if the response is bi-linear
- A DMA temperature sweep can help determine if a beta-transition will become active at high strain rates, and if high strain rate testing is needed
- More research is needed to determine the predictive equations for how to map: temperature ⇔ strain rate